tar's -t flag will list contents for you. Add that to your other flags (so -tvfz for a tar.gz, -tvfj for a tar.bz2, etc) and you can browse without extracting. From there you can extract single files quite easily

tar -xvfz mybackup.tar.gz path/to/file

The big problem with tar is remembering all the other flags. So I usually rely on 7z (of the p7zip-full package) to do all my archiving. I won't claim it is entirely better but it supports almost everything (without having to specify compression type) and the arguments are logical.

7z l archive.ext
7z e archive.ext path/to/file

It's certainly less capable, but you don't need the man page to use it.

There's also Midnight Commander (mc). This is an all-around badass for quasi-graphical terminal-based file management and with some light testing it just let you browse into both .tar.gz and .7z archives. I'm not sure how many others it supports.

Midnight Commander (mc) also has a good compressed file viewer, although I consider this a bit of cheating since mc is a file manager, albeit a text-based one.

Also, if all you want is to see what's inside compressed archives, you could learn the "view" command for each compressor. tar tzvf will show you the contents of a tar file, unzip -l will do it for a zip file, and so on.